Approximate analytical method of gain saturation analysis ofhollow waveguide lasers

Karol Janulewicz and PaweISzczepahski

A simple analytical expression is derived relating the waveguide laser output power to the distributed loss, themirror reflectances, the waveguide dimensions, and the small-signal gain coefficient. In our model thetransverse-mode distributions are taken into account and the longitudinal field distribution is approximatedby the threshold field distribution.

I. Introduction

Waveguide structures have become one of the mostcommonly used laser structures in many applications.Designing such a laser structure requires a simple for-mula to assess its output power. Almost all the previ-ous laser power calculations are based on Rigrod'smodel.' Because the exact solutions ofthis model leadto transcendental equations2 3 that are not convenientfor the design process, many approximate methodshave been developed.2 4 5 However, all these methodsare limited to the one-dimensional case neglectingtransverse field distributions in the Fabry-Perot reso-nator.

In this paper we extend the approximate methodpresented in Ref. 5 to describe nonlinear operation ofthe hollow waveguide laser including the transversefield distribution. Using the threshold field approxi-mation5 we obtain a simple approximate expressiondescribing dependence of the output power on theglobal small-signal gain coefficient, the distributedloss, the mirror reflectance, and the transverse-modedistribution. The results obtained are found to be ingood agreement with the exact solutions3 in the low-power limit. Moreover, almost all the real CO2 wave-guide lasers operate in this power limit. Thus, ourapproach can be useful in analyzing the magnitude of

The authors are with the Institute of Microelectronics and Optoe-lectronics, Warsaw University of Technology, ul. Koszykowa 75,Warsaw 00-662, Poland.

Received 14 February 1990.0003-6935/91/273818-03$05.00/0.© 1991 Optical Society of America.

mode-dependent effects on threshold gain and outputpower especially in this kind of laser.

In Section II we derive an approximate expressionrelating the output power to system parameters andpresent the laser characteristics.

II. Theory and Calculations

We analyze the laser scheme shown in Fig. 1. Theconfiguration, called dual-case I, consists of the wave-guide of the L with the distributed loss described byaimn and two mirrors with point losses aossi and reflec-tances r placed close to the ends of the waveguide.The global small-signal gain coefficient equals go.

We can transform the wave equations to the expres-sion analogous to the equation describing Rigrod'smodel:

1 djRq12ko k(

IRq|2dz I mn 1 + (IRqI + Sql2) 1Eo12/Ps amn (1)

where Rq and Sq are the forward and backward com-plex wave amplitudes of the mnth mode, respectively,E0 is the total electric field vector of the propagatingwave, ko is the free-space wave number, and flmn is thepropagating constant of the mnth mode in a wave-guide.

In our model we approximate the longitudinal fielddependence appearing in Eq. (1) by the threshold fielddistribution with unknown amplitude A. Thus, ac-cording to Ref. 5 we have

Rq(Z) = A ep(g9 0 nthz); Sq(z) = Ar2 exp[g0 ""th(2L- )], (2)

where gomnth = 1/2L ln[l/(rlr 2)]. It is easy to note that1Rq ISql = C = const. By multiplying both sides of Eq.(1) by the denominator, we can integrate the resultantequation as follows:

3818 APPLIED OPTICS / Vol. 30, No. 27 / 20 September 1991

0.014 1 .0.92 0.94 0.96 0.98 1.

ref tectance ( a. u. )Fig. 3. Amplitude small-signal gain coefficient for the lower trans-verse modes versus the transmission mirror reflectance. The re-duced output power Pout/Ps = 1.

-0.16

* .14

a,

030

C. 0.1 2

0

0.97 0.98 0.99ref lectance ( a.u. )

0.02

M~~~y

x

rR r

- Ii[>PU

E0

z: 0.017C0

M}

Z

Fig. 1. Scheme of the structure: (a) laser tube section and (b)analyzed dual-case I configuration.

0.97 0.98 0.99 1,refl ectance (a.u.

Fig. 2. Dependence of the amplitude small-signal gain coefficient

for the different transverse modes on the transmitting mirror reflec-tance Pout/Ps = 0.01.

0.1 0.96

2"mn , A [ + pE 1 (ARqY| 12c2 dz, (3)

where P1 = IRq(0)12 1Eo12 and P2 = ISq(L)12 1Eo12.For our structure (Fig. 1) we can formulate the fol-

lowing boundary conditions:

Rq(O) = Sq(O)ri, Sq(L) = Rq(L)r 2 ,

Pout = [lSq(0)l2(1 - la-(,.'2- Irl12)

+ IR(L)I2(1 - la O,2

12

- r2l2)]PT, (4)

where Pout is the output power, PT is the power propa-gating inside the resonator along the z axis, PT = SdsIETl2, and ET is the transverse component of the

Fig. 4. Reduced output power Pout/Ps vs the transmission mirrorreflectance for different transverse modes. The power gain coeffi-

cient equals 0.005 cm-'.

total electric field. Integrating Eq. (3) while takinginto account the threshold field distributions in Eqs.(2) and the boundary conditions in Eqs. (4), after somecalculations we can write Eq. (3) in the following form:

Tu 2L(go - a.,,) -o ln(l/rlr2)] P ., 0 mu mn n r 1 2

(1-lal -12_lr1 2)/rl + (1 - lalO.5212 - r2 )/r2

X , (5)

I + ko n(/r r W)J(1/rl + 1/r2)(1 - rlr2)

20 September 1991 / Vol. 30, No. 27 / APPLIED OPTICS 3819

(a)

rst_ Rq(Z)

Cb)

0.0008

E

- 0.00050)

cM

0.0002 '0.96

JP2 1Pi lIRql2

IEOI2/,+ P 11

Ps k + C )]dIRq12 = 2 ° J 0godk

1.

where the normalization factor P, is Pv = f dsIEoI2.Equation (5) can be rewritten to express the globalsmall-signal gain as a function of the output power andthe system parameters as follows:

go = C"mn +an 1l n(llrlr2) +fn 2kom PVb kamn 2L ~I~2.2kJ P P,

(11r, + 1/r2)(1 - rr2)[1 + ko ,m 2L/ln(l/rlr2)]

[(1-laloxi 2-_rl2)1rj][(j-la .212-_r212)] (6)

In Eqs. (5) and (6) Pout = PoudSw is the output powerdensity averaged over the waveguide cross section.

The laser characteristics of the reduced output pow-er density Pout/PS and the amplitude global small-signal gain coefficient obtained with the help of Eqs.(5) and (6) are presented in Figs. 2-4. The character-istics are obtained for three transverse modes, ExH11,ExH12, and ExH13. The calculations are carried out for2a = 2b = 1.5 mm, aloss' = aoss2 = 0, and the indices ofthe waveguide walls equal nAl = 20- j59 and nalumina =0.65 - j0.038, respectively. The laser length is L = 32cm.

It is worth noting that the results obtained with thehelp of the simple approximate relations, Eqs. (5) and(6), are found to be in qualitative agreement with exactsolutions5 and the solutions obtained in Ref. 6. More-over, as we can see in Figs. 2-4 for each transversemode for a given output power level an optimum out-put mirror reflectivity exists that results in the mini-mum global small-signal gain required to maintain theoutput power level. The optimum mirror reflectivityis shifted toward smaller values by increasing the modenumber.

Ill. Conclusions

We have presented a simple way to estimate the mag-nitude of mode-dependent effects on threshold gain

and output power. Comparison of the results in thispaper with those in Ref. 6 implies that knowledge ofthe real transverse-mode distribution is necessary toobtain more real-gain saturation analysis of hollowwaveguide lasers. It is obvious that Eqs. (5) and (6)are the nearly plane-wave solutions and give underes-timated results in comparison with the typical valuesof the small-signal gain used in practice.7 On the otherhand, the solutions for the pure transverse-mode dis-tributions6 give an overestimated value. Thus, we cansay that the real transverse-mode structures in thehollow waveguide lasers are a combination of puretransverse modes that can be excited, for example, bywaveguide inhomogeneities. Thus, the resultant fielddistributions are intermediate ones between the fieldof a plane wave and the transverse waveguide mode.This conclusion confirms the results of measurementsand suggestions presented by Hill7 on the complexityof the real mode structure in such lasers.

References

1. W. W. Rigrod, "Saturation effects in high-gain lasers," J. Appl.Phys. 36, 2487-2490 (1965).

2. W. W. Rigrod, "Homogeneously broadened cw lasers with uni-form distributed loss," IEEE J. Quantum Electron. QE-14, 377-381 (1978).

3. G. Schindler, "Optimum output efficiency of homogeneouslybroadened lasers with constant loss," IEEE J. Quantum Electron.QE-15, 546-549 (1980).

4. D. Eimerl, "Optical extraction characteristics of homogeneouslybroadened cw lasers with nonsaturable losses," J. Appl. Phys. 51,3006 (1980).

5. A. Kujawski and P. Szczepanski, "Gain saturation in hollowwaveguide lasers," J. Mod. Opt. (to be published).

6. K. Janulewicz and P. Szczepaiiski, paper to be published.7. C. A. Hill, "Transverse modes of plane-mirror waveguide resona-

tors," IEEE J. Quantum Electron. 24, 1936-1946 (1988).

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